For this week of Pre-Calculus 11, I learned about how to use the Quadratic Formula to solve quadratic equations.
In math, using a formula often makes equations very easy to solve, as seen with the Pythagorean Theorem, or even your usual area-of-a-shape formula. For the Quadratic Formula, it makes use of our ax squared plus bx + c = 0 quadratic equation, where one would assign certain roles to a, b, and c in the formula.
In order to solve a quadratic equation, the Quadratic Formula, firstly, requires that the equation equals to 0. Secondly, the formula must equal to 2 solutions or ‘2 same’ solutions, noted by our ‘+ or -‘ sign between –b and the square root. But sometimes, there can be quadratic equations that do not have any solution. Our discriminant (also known as the radicand in this formula, but without the square root sign), will determine whether our equation will have 2 solutions, 1 solution (two solutions that equal the same, like 0), or no real solutions at all.
To find the amount of solutions, I need to look at whether the discriminant, as a whole, will be rational or irrational, and positive or negative. Take a look at the following 4 examples.
For the first example, I can see that the discriminant will equal to 0, a rational number that, when put in a quadratic formula, will produce only 1 solution. This is because we can do a square root of 0 and it would still be 0, and a number added or subtracted by 0 will always be the same; therefore, this quadratic equation has only 1 solution. Nothing else can change x from being -2.
The second example instead equals to a natural number. 36 would also give a rational answer if it was square rooted, which means that 36 is a perfect square. This means, that if we were to put 36 as our discriminant in a Quadratic Formula, we would get a rational solution. But there’s an easier way to find our 2 solutions when we get a perfect square as a discriminant: just solve by factoring!
The third example, unlike the previous ones, is instead an irrational discriminant. However, this will still give us 2 solutions because it is positive. We will just have a mixed radical instead of a whole number in our Quadratic Formula.
The last example, because the discriminant is a negative, will not generate any solutions. This is due to how we cannot find a square root of a negative number. There are no real numbers that, multiplied by itself, equal a negative number.
Sometimes we might not even have a value for a, b, or c. If that is the case, do not worry at all! Just continue on with the formula as it is – replace the variables that do not have values with 0.
Using the Quadratic Formula and its discriminant can really help speed the process in finding a solution for some quadratic equations. While the formula is more of a last-resort compared to the factoring and completing-the-square strategies, it can be quite useful when dealing with an equation that has irrational solutions, or if you just want to make sure it’s solvable.